| L(s) = 1 | + 2-s − 3.13·3-s + 4-s + 2.17·5-s − 3.13·6-s + 2.94·7-s + 8-s + 6.84·9-s + 2.17·10-s − 0.677·11-s − 3.13·12-s − 1.92·13-s + 2.94·14-s − 6.83·15-s + 16-s − 0.0568·17-s + 6.84·18-s − 19-s + 2.17·20-s − 9.22·21-s − 0.677·22-s − 7.41·23-s − 3.13·24-s − 0.261·25-s − 1.92·26-s − 12.0·27-s + 2.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.973·5-s − 1.28·6-s + 1.11·7-s + 0.353·8-s + 2.28·9-s + 0.688·10-s − 0.204·11-s − 0.905·12-s − 0.534·13-s + 0.786·14-s − 1.76·15-s + 0.250·16-s − 0.0137·17-s + 1.61·18-s − 0.229·19-s + 0.486·20-s − 2.01·21-s − 0.144·22-s − 1.54·23-s − 0.640·24-s − 0.0523·25-s − 0.377·26-s − 2.32·27-s + 0.555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369531354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369531354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 0.677T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 + 0.0568T + 17T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 3.10T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 0.0900T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 - 9.62T + 53T^{2} \) |
| 59 | \( 1 + 2.91T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71373013696877467389656504166, −6.65214919109735274837763074426, −6.34376918947717939837451986708, −5.62819745116442886560954722488, −5.13087782445306375391718152026, −4.62608872017286268986914858394, −3.93348528730545236618366425244, −2.39822836089079339501079795827, −1.78228583780298795432815565292, −0.76903881227515801820375499648,
0.76903881227515801820375499648, 1.78228583780298795432815565292, 2.39822836089079339501079795827, 3.93348528730545236618366425244, 4.62608872017286268986914858394, 5.13087782445306375391718152026, 5.62819745116442886560954722488, 6.34376918947717939837451986708, 6.65214919109735274837763074426, 7.71373013696877467389656504166
